## Abstract

A method is developed for interpreting the statistics of the sun’s glitter on the sea surface in terms of the statistics of the slope distribution. The method consists of two principal phases: (1) of identifying, from geometric considerations, any point on the surface with the particular slope required for the reflection of the” sun’s rays toward the observer; and (2) of interpreting the average brightness of the sea surface in the vicinity of this point in terms of the frequency with which this particular slope occurs. The computation of the probability of large (and infrequent) slopes is limited by the disappearance of the glitter into a background consisting of (1) the sunlight scattered from particles beneath the sea surface, and (2) the skylight reflected by the sea surface.

The method has been applied to aerial photographs taken under carefully chosen conditions in the Hawaiian area. Winds were measured from a vessel at the time and place of the aerial photographs, and cover a range from 1 to 14 m sec^{−1}. The effect of surface slicks, laid by the vessel, are included in the study. A two-dimensional Gram-Charlier series is fitted to the data. As a first approximation the distribution is Gaussian and isotropic with respect to direction. The mean square slope (regardless of direction) increases linearly with the wind speed, reaching a value of (tan16°)^{2} for a wind speed of 14 m sec^{−1}. The ratio of the up/ downwind to the crosswind component of mean square slope varies from 1.0 to 1.9. There is some up/downwind skewness which increases with increasing wind speed. As a result the most probable slope at high winds is not zero but a few degrees, with the azimuth of ascent pointing downwind. The measured peakedness which is barely above the limit of observational error, is such as to make the probability of very large and very small slopes greater than Gaussian. The effect of oil slicks covering an area of one-quarter square mile is to reduce the mean square slopes by a factor of two or three, to eliminate skewness, but to leave peakedness unchanged.

© 1954 Optical Society of America

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### Equations (36)

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(1)
$$\begin{array}{cc}{z}_{x}=\partial z/\partial x=sin\alpha tan\beta ,& {z}_{y}=\partial z/\partial y=cos\alpha tan\beta ,\end{array}$$
(2)
$$\begin{array}{lll}{a}_{n}=-sin\alpha sin\beta ,\hfill & {b}_{n}=-cos\alpha sin\beta ,\hfill & {c}_{n}=cos\beta .\hfill \end{array}$$
(3)
$$\begin{array}{lll}{a}_{i}=0,\hfill & {b}_{i}=-cos\varphi ,\hfill & {c}_{i}=-sin\varphi ,\hfill \end{array}$$
(4)
$$\begin{array}{lll}{a}_{r}=-sin\nu sin\mu ,\hfill & {b}_{r}=-cos\nu sin\mu ,\hfill & {c}_{r}=cos\mu .\hfill \end{array}$$
(5)
$$\begin{array}{c}\begin{array}{c}\begin{array}{cc}{a}_{r}-{a}_{i}=2{a}_{n}cos\omega ,& {b}_{r}-{b}_{i}=2{b}_{n}cos\omega ,\end{array}\\ {c}_{r}-{c}_{i}=2{c}_{n}cos\omega ,\end{array}\end{array}$$
(6)
$$cos\omega =cos\beta sin\varphi -cos\alpha sin\hspace{0.17em}\beta cos\varphi .$$
(7)
$$\begin{array}{ll}cos\mu \hfill & =2cos\beta cos\omega -sin\varphi ,\hfill \\ cot\nu \hfill & =cot\alpha -\frac{1}{2}csc\alpha csc\beta sec\omega cos\varphi .\hfill \end{array}$$
(8)
$${\mathrm{\Delta}}_{t}=\frac{1}{4}\pi {\u220a}^{2}{sec}^{3}{\beta}_{0}sec{\omega}_{0},$$
(9)
$$P({z}_{x0},{z}_{y0})=p({z}_{x0},{z}_{y0}){\mathrm{\Delta}}_{t}({z}_{x0},{z}_{y0}).$$
(10)
$$J=\rho (\omega ){\mathrm{\Delta}}_{h}H{(\pi {\u220a}^{2})}^{-1}sec\beta cos\omega $$
(12)
$$p=4{\rho}^{-1}({cos}^{4}\beta )Ncos\mu /H.$$
(13)
$$2\rho (\omega )={sin}^{2}(\omega -{\omega}^{\prime}){csc}^{2}(\omega +{\omega}^{\prime})+{tan}^{2}(\omega -{\omega}^{\prime}){cot}^{2}(\omega -{\omega}^{\prime}),$$
(14)
$$E=\frac{1}{4}\pi {d}^{2}{s}^{-2}\tau (\mathrm{\lambda})t{\mathrm{\Delta}}_{u}cos\mathrm{\lambda}cos\mu $$
(15)
$$\begin{array}{ll}N\hfill & =C{sec}^{4}\mathrm{\lambda}{(\tau )}^{-1}E(D),\hfill \\ C\hfill & =(4{f}^{2})/(\pi {d}^{2}t).\hfill \end{array}$$
(16)
$${N}^{\prime}=sec\mu \mathit{\int}\mathit{\int}{N}_{s}\rho (\omega )cos\omega sec\beta p({z}_{x},{z}_{y})d{z}_{x}d{z}_{y},$$
(17)
$$cos\omega =cos\beta (cos\mu +{{z}_{y}}^{\prime}sin\mu ).$$
(18)
$${N}^{\prime}={(\pi {\sigma}^{2})}^{-1}{N}_{s}\rho (\mu )\mathit{\int}\mathit{\int}(1+{{az}_{y}}^{\prime}+{{bz}_{y}}^{\prime 2}+{{cz}_{x}}^{\prime 2}+\cdots )\times exp[-({{z}_{x}}^{\prime 2}+{{z}_{y}}^{\prime 2})/{\sigma}^{2}]{{dz}_{x}}^{\prime}{{dz}_{y}}^{\prime},$$
(19)
$$\begin{array}{lll}a=-({F}^{\prime}/F),\hfill & b=\frac{1}{2}+\frac{1}{2}({F}^{\u2033}/F),\hfill & c=\frac{1}{2}+\frac{1}{2}({F}^{\prime}/F)cot\mu \hfill \end{array}.$$
(20)
$$k={\sigma}^{-1}cot\mu .$$
(21)
$${N}^{\prime}={N}_{s}\rho (\mu )\{\frac{1}{2}[1+I(k)]+\frac{1}{2}{\pi}^{-\frac{1}{2}}a\sigma {e}^{-{k}^{2}}+\frac{1}{4}b{\sigma}^{2}[1+I(k)-2{\pi}^{-\frac{1}{2}}k{e}^{-{k}^{2}}]+\frac{1}{4}c{\sigma}^{2}[1+I(k)]+\cdots \},$$
(22)
$$I(k)=2{\pi}^{-\frac{1}{2}}{\mathit{\int}}_{0}^{k}{e}^{-{t}^{2}}dt$$
(23)
$$k={(2\sigma )}^{-1}cot\mu $$
(24)
$$p=[\text{two}\u2010\text{dimensional Gaussian distribution}]\times [1+\text{\u2211}_{i,j=1}^{\infty}{c}_{ij}{H}_{i}(\xi ){H}_{j}(\eta )].$$
(25)
$$\begin{array}{ll}\xi ={{z}_{x}}^{\prime}/{\sigma}_{c},\hfill & \eta ={{z}_{y}}^{\prime}/{\sigma}_{u},\hfill \end{array}$$
(26)
$$\begin{array}{ll}p({{z}_{x}}^{\prime},{{z}_{y}}^{\prime})\hfill & ={(2\pi {\sigma}_{c}{\sigma}_{u})}^{-1}exp-\frac{1}{2}({\xi}^{2}+{\eta}^{2})\hfill \\ \hfill & \times [1-\frac{1}{2}{c}_{21}({\xi}^{2}-1)\eta -\frac{1}{6}{c}_{03}({\eta}^{3}-3\eta )\hfill \\ \hfill & +(1/24){c}_{40}({\xi}^{4}-6{\xi}^{2}+3)\hfill \\ \hfill & +\frac{1}{4}{c}_{22}({\xi}^{2}-1)({\eta}^{2}-1)+(1/24){c}_{04}({\eta}^{4}-6{\eta}^{2}+3)+\cdots ].\hfill \end{array}$$
(27)
$$\begin{array}{ll}{{\sigma}_{c}}^{2}\hfill & \begin{array}{cc}=0.003+1.92\times {10}^{-3}W\pm 0.002& r=0.956\end{array}\hfill \\ {{\sigma}_{u}}^{2}\hfill & \begin{array}{cc}=0.000+3.16\times {10}^{-3}W\pm 0.004& r=0.945\end{array}\hfill \\ {{\sigma}_{c}}^{2}+{{\sigma}_{u}}^{2}\hfill & \begin{array}{cc}=0.003+5.12\times {10}^{-3}W\pm 0.004& r=0.986\end{array}\hfill \end{array}$$
(28)
$$\begin{array}{ll}{{\sigma}_{c}}^{2}\hfill & \begin{array}{cc}=0.003+1.84\times {10}^{-3}W\pm 0.002& r=0.78\end{array}\hfill \\ {{\sigma}_{u}}^{2}\hfill & \begin{array}{cc}=0.005+0.78\times {10}^{-3}W\pm 0.002& r=0.70\end{array}\hfill \\ {{\sigma}_{c}}^{2}+{{\sigma}_{u}}^{2}\hfill & \begin{array}{cc}=0.008+1.56\times {10}^{-3}W\pm 0.004& r=0.77\end{array}\hfill \end{array}$$
(29)
$${c}_{21}=0.01-0.0086W\pm 0.03$$
(30)
$${c}_{03}=0.04-0.033W\pm 0.12.$$
(31)
$$\begin{array}{lll}{c}_{21}=0.00\pm 0.02\hfill & \text{and}\hfill & {c}_{03}=0.02\pm 0.05.\hfill \end{array}$$
(32)
$$\begin{array}{lll}{c}_{40}=0.40\pm 0.23\hfill & {c}_{22}=0.12\pm 0.06\hfill & {c}_{04}=0.23\pm 0.41\hfill \end{array}$$
(33)
$$\begin{array}{lll}0.36\pm 0.24\hfill & 0.10\pm 0.05\hfill & 0.26\pm 0.31\hfill \end{array}$$
(34)
$$\begin{array}{llll}pd{{z}_{x}}^{\prime}d{{z}_{y}}^{\prime},\hfill & mpd{\alpha}^{\prime}dm,\hfill & \text{and}\hfill & ptan\beta {sec}^{2}\beta d{\alpha}^{\prime}d\beta \hfill \end{array}$$
(35)
$$\begin{array}{llll}{{z}_{x}}^{\prime}+\frac{1}{2}d{{z}_{x}}^{\prime},\hfill & {{z}_{y}}^{\prime}\pm \frac{1}{2}d{{z}_{y}}^{\prime};\hfill & {\alpha}^{\prime}\pm \frac{1}{2}d{\alpha}^{\prime},\hfill & m\pm \frac{1}{2}dm;\hfill \end{array}$$
(36)
$$\begin{array}{ll}{\alpha}^{\prime}\pm \frac{1}{2}d{\alpha}^{\prime},\hfill & \beta \pm \frac{1}{2}d\beta ,\hfill \end{array}$$